Based on the given picture, it’s known that:
Hence:
TQ = QS × sin(∠QST)⇒ TQ = 25 × sin(60°)⇒ TQ = 25 × ½√3
PT = TQ / tan(∠QPT)⇒ PT = TQ / tan(90°–(90°–∠SQT))⇒ PT = TQ / tan(∠SQT)⇒ PT = TQ / tan(90°–∠QST)⇒ PT = TQ / tan(90°–60°)⇒ PT = TQ / tan(30°) ∴ [ ∠QPT = 30° ]⇒ PT = TQ × cot(30°)⇒ PT = (25√3)/2 × cos(30°) / sin(30°)⇒ PT = (25√3)/2 × ½√3 / ½⇒ PT = (25√3)/2 × √3⇒ PT = 75/2 cm
PR = PQ + QR⇒ PR = PQ + QS [ because ∠QSR = ∠QRS = 45° ]⇒ PR = [TQ / sin(∠QPT)] + QS⇒ PR = [(25√3)/2 / sin(30°)] + 25⇒ PR = [(25√3)/2 / ½)] + 25⇒ PR = [(25√3)/2 × 2)] + 25
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Explanation
Right-Angled Triangle (Right Triangle)
Based on the given picture, it’s known that:
Hence:
TQ = QS × sin(∠QST)
![\therefore\ \boxed{\ \bf TQ=\frac{25\sqrt{3}}{2}\ cm\ } \therefore\ \boxed{\ \bf TQ=\frac{25\sqrt{3}}{2}\ cm\ }](https://tex.z-dn.net/?f=%5Ctherefore%5C%20%5Cboxed%7B%5C%20%5Cbf%20TQ%3D%5Cfrac%7B25%5Csqrt%7B3%7D%7D%7B2%7D%5C%20cm%5C%20%7D)
⇒ TQ = 25 × sin(60°)
⇒ TQ = 25 × ½√3
PT = TQ / tan(∠QPT)
![\therefore\ \boxed{\ \bf PT=37.5\ cm\ } \therefore\ \boxed{\ \bf PT=37.5\ cm\ }](https://tex.z-dn.net/?f=%5Ctherefore%5C%20%5Cboxed%7B%5C%20%5Cbf%20PT%3D37.5%5C%20cm%5C%20%7D)
⇒ PT = TQ / tan(90°–(90°–∠SQT))
⇒ PT = TQ / tan(∠SQT)
⇒ PT = TQ / tan(90°–∠QST)
⇒ PT = TQ / tan(90°–60°)
⇒ PT = TQ / tan(30°) ∴ [ ∠QPT = 30° ]
⇒ PT = TQ × cot(30°)
⇒ PT = (25√3)/2 × cos(30°) / sin(30°)
⇒ PT = (25√3)/2 × ½√3 / ½
⇒ PT = (25√3)/2 × √3
⇒ PT = 75/2 cm
PR = PQ + QR
![\therefore\ \boxed{\ \begin{aligned}\bf PR&\bf=25+25\sqrt{3}\ cm\\&\bf=25\left(1+\sqrt{3}\right)\ cm\end{aligned}\ } \therefore\ \boxed{\ \begin{aligned}\bf PR&\bf=25+25\sqrt{3}\ cm\\&\bf=25\left(1+\sqrt{3}\right)\ cm\end{aligned}\ }](https://tex.z-dn.net/?f=%5Ctherefore%5C%20%5Cboxed%7B%5C%20%5Cbegin%7Baligned%7D%5Cbf%20PR%26%5Cbf%3D25%2B25%5Csqrt%7B3%7D%5C%20cm%5C%5C%26%5Cbf%3D25%5Cleft%281%2B%5Csqrt%7B3%7D%5Cright%29%5C%20cm%5Cend%7Baligned%7D%5C%20%7D)
⇒ PR = PQ + QS [ because ∠QSR = ∠QRS = 45° ]
⇒ PR = [TQ / sin(∠QPT)] + QS
⇒ PR = [(25√3)/2 / sin(30°)] + 25
⇒ PR = [(25√3)/2 / ½)] + 25
⇒ PR = [(25√3)/2 × 2)] + 25